Question: The length, width, and height of a rectangular prism are in the ratio $3:2:1$, respectively. What is the length of the prism if it has a volume of $48\text{ in}^3$?
Answer: Let $l$, $w$, and $h$ represent the length, width, and height of the prism, respectively. Thus, from the given ratio, we know that $l=3h$ and $w=2h$, such that $lwh=48$. Since we know $l$ and $w$ in terms of $h$, we have $$48=lwh=(3h)(2h)h=6h^3$$So $h^3=48/6=8$ and $h=2$. We can then solve for $l$ to get $l=3\cdot 2 = 6$.